Spectral properties of birth-death polynomials

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Journal of Computational and Applied

Mathematics

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Spectral properties of birth–death polynomials

Erik A. van Doorn

∗

Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

a r t i c l e i n f o

Article history:

Received 30 March 2014

Received in revised form 4 June 2014

MSC: primary 42C05 secondary 60J80 Keywords: Birth–death process Orthogonal polynomials Orthogonalizing measure Spectrum

Stieltjes moment problem

a b s t r a c t

We consider sequences of polynomials that are defined by a three-terms recurrence relation and orthogonal with respect to a positive measure on the nonnegative axis. By a famous result of Karlin and McGregor such sequences are instrumental in the analysis of birth–death processes. Inspired by problems and results in this stochastic setting we present necessary and sufficient conditions in terms of the parameters in the recurrence relation for the smallest or second smallest point in the support of the orthogonalizing measure to be larger than zero, and for the support to be discrete with no finite limit point.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

We are concerned with a sequence of polynomials

{

Pn

}

defined by the three-terms recurrence relation

Pn+1

(

x

) = (

x

−

λ

n

−

µ

n

)

Pn

(

x

) − λ

n−1

µ

nPn−1

(

x

),

n

>

0

,

P1

(

x

) =

x

−

λ

0

−

µ

0

,

P0

(

x

) =

1

,

(1) where

λ

n

>

0 for n

≥

0

, µ

n

>

0 for n

≥

1 and

µ

0

≥

0. Since polynomial sequences of this type play an important role in the

analysis of birth–death processes – continuous-time Markov chains on an ordered set with transitions only to neighbouring states – we will refer to

{

Pn

}

as the sequence of birth–death polynomials associated with the birth rates

λ

nand death rates

µ

n. For more information on the relation between a sequence of birth–death polynomials and the corresponding birth–death process we refer to the seminal papers of Karlin and McGregor [1,2].

By Favard’s theorem (see, for example, Chihara [3]) there exists a probability measure (a Borel measure of total mass 1) on R with respect to which the polynomials Pnare orthogonal. In the terminology of the theory of moments the Hamburger

moment problem associated with the polynomials Pn is solvable. Actually, as shown by Karlin and McGregor [1] and Chihara [4] (see also [3, Theorem I.9.1 and Corollary]), even the Stieltjes moment problem associated with

{

Pn

}

is solvable, which means that there exists an orthogonalizing measure

ψ

for

{

Pn

}

with support on the nonnegative axis, that is,



[0,∞)

Pn

(

x

)

Pm

(

x

)ψ(

dx

) =

kn

δ

nm

,

n

,

m

≥

0

,

(2) with kn

>

0. The Stieltjes moment problem associated with

{

Pn

}

is said to be determined if

ψ

is uniquely determined by (2), and indeterminate otherwise. In the latter case there is, by [5, Theorem 5], a unique orthogonalizing measure for which

∗_{Tel.: +31 53 4893387.}

E-mail addresses:e.a.vandoorn@utwente.nl,e.a.vandoorn@hotmail.com.

http://dx.doi.org/10.1016/j.cam.2014.08.014

the infimum of its support is maximal. We will refer to this measure as the natural measure for

{

Pn

}

. In what follows

ψ

will always refer to the natural measure for

{

Pn

}

if the Stieltjes moment problem associated with

{

Pn

}

is indeterminate.

Of particular interest to us are the quantities

ξ

i, recurrently defined by

ξ

1

:=

inf supp

(ψ),

(3)

and

ξ

i+1

:=

inf

{

supp

(ψ) ∩ (ξ

i

, ∞)},

i

≥

1

,

(4)

where supp

(ψ)

denotes the support of the measure

ψ

, also referred to as the spectrum of

ψ

(or of the polynomials Pn). In addition we let

σ :=

lim

i→∞

ξ

i

,

(5)

the first limit point of supp

(ψ)

if it exists, and infinity otherwise. It is clear from the definition of

ξ

ithat, for all i

≥

1,

ξ

i+1

≥

ξ

i

≥

0

,

and

ξ

i

=

ξ

i+1

⇐⇒

ξ

i

=

σ .

In the analysis of a birth–death process on a countable state space – a birth–death process on the nonnegative integers with birth rate

λ

nand death rate

µ

nin state n, say – the question of whether the time-dependent transition probabilities of the process converge to their limiting values exponentially fast as time goes to infinity has attracted considerable attention. This question may be translated into the setting of the polynomials Pnof(1)by asking whether

ξ

1

>

0, and if not, whether

ξ

2

>

0, since the exponential rate of convergence (or decay parameter)

α

of the birth–death process satisfies

α =



ξ

1 if

ξ

1

>

0

ξ

2 if

ξ

2

> ξ

1

=

0

0 if

ξ

2

=

ξ

1

=

0

(see, for example, [6]). Note that

α >

0

⇐⇒

0

< σ ≤ ∞,

(6)

so the above question may be rephrased by asking whether 0

< σ ≤ ∞

. Recent results, in particular in the Chinese literature, have culminated in a complete solution of the problem in the stochastic setting by revealing simple and easily verifiable conditions for exponential convergence in terms of the birth and death rates. The purpose of this paper is to present these results in an orthogonal-polynomial context, and to provide new proofs for some of the results by using tools from the orthogonal-polynomial toolbox. Our methods enable us also to establish a simple, necessary and sufficient condition for

σ = ∞

, that is, for the spectrum of the orthogonalizing measure to be discrete with no finite limit point, thus extending another recent result.

Before stating the results in Section3and discussing proofs in Section4we present a number of preliminary results in Section2. Additional information on related literature and some concluding remarks will be given in Section5.

2. Preliminaries

It will be convenient to define the constants

π

nby

π

0

:=

1 and

π

n

:=

λ

0

λ

1

. . . λ

n−1

µ

1

µ

2

. . . µ

n

,

n

>

0

,

(7)

and to use the shorthand notation

Kn

:=

n



i=0

π

i

,

n

≥

0

,

K∞

:=

∞



i=0

π

i

,

(8) and Ln

:=

n



i=0

(λ

i

π

i

)

−1

_,

n

≥

0

,

L∞

:=

∞



i=0

(λ

i

π

i

)

−1

_.

(9) With the convention that the measure

ψ

in(2)is interpreted as the natural measure if the Stieltjes moment problem associated with

{

Pn

}

is indeterminate, the quantities

ξ

iand

σ

of(3)–(5)may be defined alternatively in terms of the (simple and positive) zeros of the polynomials Pn

(

x

)

(see [3, Section II.4]). Namely, with xn1

<

xn2

< · · · <

xnndenoting the n zeros of Pn

(

x

)

, we have the classic separation result

so that the limits as n

→ ∞

of xniexist, while lim

n→∞xni

=

ξ

_i

,

i

=

1

,

2

, . . . .

If the Stieltjes moment problem associated with

{

Pn

}

is indeterminate then, by [5, Theorems 4 and 5], we have

ξ

i+1

> ξ

i

>

0 for all i

≥

1 and

σ =

limi→∞

ξ

i

= ∞

, so that the spectrum of the (natural) measure

ψ

actually coincides with the set

{

ξ

₁

, ξ

₂

, . . .}

. So in this setting the questions of whether

ξ

1

>

0 and the spectrum is discrete with no finite limit point can be

answered in the affirmative. It is therefore no restriction to assume in what follows that

K∞

+

L∞

= ∞

,

(10)

which, by [7, Theorem 2], is necessary – and, if

µ

0

=

0, also sufficient – for the Stieltjes moment problem associated with

{

Pn

}

to be determined.

Under these circumstances we know from [2] (or from classic results on the moment problem in [8]) that

ψ({

0

}

) =



₁ K∞ if

µ

0

=

0 and K∞

< ∞

0 otherwise

,

(11) so that

µ

0

>

0 or

(µ

0

=

0 and L∞

< ∞) H⇒ ξ

1

>

0 or

σ =

0

.

(12)

Actually, under the premise in(12)the measure

ψ

has a finite moment of order

−

1, since, by [2, (9.9) and (9.14)],



∞ 0

ψ(

dx

)

x

=

L∞ 1

+

µ

₀L∞

,

(13) which, if L∞

= ∞

, should be interpreted as infinity if

µ

0

=

0 and as

µ

−01if

µ

0

>

0.

3. Results

In what follows we maintain the assumption K∞

+

L∞

= ∞

. Our first proposition deals with a simple case.

Proposition 1. If K∞

=

L∞

= ∞

then

σ =

0.

Indeed, for

µ

0

=

0 this result follows immediately from(11)and(13), while it is known (see [6, p. 527]) that changing the

value of

µ

0(or, for that matter, of any finite number of birth and death rates) does not affect the value of

σ

.

Our next result is a proposition on the basis of which all the remaining results of this section can be obtained using orthogonal-polynomial techniques.

Proposition 2. Let K∞

< ∞

and

µ

0

>

0. Then

1

4R

≤

ξ

1

≤

1

R

if R

:=

sup_nLn

(

K∞

−

Kn

) < ∞

, and

ξ

1

=

0 otherwise.

This proposition was stated explicitly for the first time (in terms of the decay parameter of an absorbing birth–death process) by Sirl et al. [9]. These authors do not provide a proof, but note that the techniques employed by Chen to analyse ergodic birth–death processes – which in our setting correspond to the case K∞

< ∞

and

µ

0

=

0 – are applicable to absorbing

birth–death processes as well (see in particular [10, Theorem 3.5]). Mu-Fa Chen himself stated the result of Proposition 2 explicitly in [11, Theorem 4.2]. Chen’s technique involves Dirichlet forms, but recently Proposition 2 was proven in [12] using orthogonal-polynomial and eigenvalue techniques. A sketch of the argument employed in [12], emphasizing and elucidating the orthogonal-polynomial aspects, will be given in Section4.

We next list a number of results as corollaries of thePropositions 1and2.

Corollary 1. (i) If K∞

< ∞

and

µ

0

>

0, then

ξ

1

>

0

⇐⇒

0

< σ ≤ ∞ ⇐⇒

sup

n

Ln

(

K∞

−

Kn

) < ∞.

(ii) If K∞

< ∞

and

µ

0

=

0, then

ξ

1

=

0 and

ξ

2

>

0

⇐⇒

0

< σ ≤ ∞ ⇐⇒

sup n Ln

(

K∞

−

Kn

) < ∞.

(iii) If L∞

< ∞

, then

ξ

1

>

0

⇐⇒

0

< σ ≤ ∞ ⇐⇒

sup n Kn

(

L∞

−

Ln−1

) < ∞.

Corollary 2. If

σ = ∞

then K∞

< ∞

or L∞

< ∞

. Moreover, (i) if K∞

< ∞

, then

σ = ∞ ⇐⇒

lim n→∞Ln

(

K∞

−

Kn

) =

0

;

(ii) if L∞

< ∞

, then

σ = ∞ ⇐⇒

lim n→∞Kn

(

L∞

−

Ln−1

) =

0

.

Corollary 1(i) is [9, Corollary 1].Corollary 1(ii) (in the setting of birth–death processes) is the oldest result and was first presented by Mu-Fa Chen in [10]. Together with many related and more refined results, the statements (i) and (iii) of Corollary 1appear in the survey paper [11].Corollary 2(i) for the case

µ

0

=

0 was presented by Mao in [13], but announced

already as a result of Mao’s in [14]. In its generalityCorollary 2is new.

4. Proofs

Obviously,Corollary 1(i) follows immediately from(12)andProposition 2, and the first statement ofCorollary 2from Proposition 1. The proofs of the remaining statements in theCorollaries 1and2will be given in three steps. In the first step, elaborated in Section4.1, we will show that by employing the duality concept for birth–death processes introduced by Karlin and McGregor [1,2] one can show that the results of both corollaries for the case L∞

< ∞

are implied by the results for the case K∞

< ∞

.

In the second step, elaborated in Section4.2, we will show that by using properties of co-recursive polynomials the statements of the corollaries for the case K∞

< ∞

and

µ

0

=

0 are implied by the results for the case K∞

< ∞

and

µ

0

>

0.

In Section4.3we will apply results on associated polynomials to obtain the statement ofCorollary 2for the case K∞

< ∞

and

µ

0

>

0 fromCorollary 1(i). As announced, we conclude in Section4.4with a sketch of the proof of Proposition 2

presented in [12], and some elucidative remarks.

4.1. Dual polynomials

Our point of departure in this subsection is a sequence of birth–death polynomials

{

Pn

}

satisfying the recurrence relation (1)with

µ

0

>

0. Following Karlin and McGregor [1,2], we define the dual polynomials Pndby a recurrence relation similar to (1)but with parameters

λ

d

nand

µ

dngiven by

µ

d0

=

0 and

λ

d

n

:=

µ

n

,

µ

dn+1

:=

λ

n

,

n

≥

0

.

Accordingly, we define

π

₀d

=

1 and, for n

≥

1,

π

d n

=

λ

d 0

λ

d1

. . . λ

dn−1

µ

d 1

µ

d 2

. . . µ

dn

=

µ

0

µ

1

. . . µ

n−1

λ

0

λ

1

. . . λ

n−1

,

and note that

π

d n+1

=

µ

0

(λ

n

π

n

)

−1 and

(λ

dn

π

d n

)

−1

₌

_µ

−1 0

π

n

.

(14)

So the assumption(10)is equivalent to ∞



n=0



π

d n

+

(λ

dn

π

nd

)

−1

_{ = ∞}

_.

The polynomials Pnand Pndare easily seen to be related by

P_nd+1

(

x

) =

Pn+1

(

x

) + λ

nPn

(

x

),

n

≥

0

.

(15) In the terminology of Chihara [3, Section I.7–9] the polynomials Pn are the kernel polynomials (with

κ

-parameter 0) corresponding to the polynomials P_nd. As a consequence, there is a unique (natural) measure

ψ

don the nonnegative real axis with respect to which the polynomials Pd

nare orthogonal. By [1, Lemma 3] we actually have

µ

0

ψ([

0

,

x

]

) =

x

ψ

d

([

0

,

x

]

),

x

≥

0

.

With

ξ

d

i and

σ

ddenoting the quantities defined by(3)–(5)if we replace

ψ

by

ψ

d, we thus have, for i

≥

1,

ξ

i

=

ξ

d i+1 if

ξ

d 1

=

0 and

σ

d

_>

₀

ξ

d i otherwise

,

(16) and

σ

d

₌

_{σ .}

₍₁₇₎

With(14)and(16)it is now easy to see that statement (iii) ofCorollary 1is implied by statement (ii) if

µ

0

>

0, and by

4.2. Co-recursive polynomials

Our point of departure in this subsection is the sequence of birth–death polynomials

{

Pn

}

satisfying the recurrence relation(1)with

µ

0

=

0. With

{

Pn

}

we associate a sequence of birth–death polynomials

{

Pn∗

}

with parameters

λ

∗ n and

µ

∗

nthat are identical to those of

{

Pn

}

except that

µ

∗0

=

c

>

0. So the polynomials P

∗ nsatisfy P ∗ 0

(

x

) =

1 and P_n∗+1

(

x

) = (

x

−

λ

n

−

µ

n

)

P ∗ n

(

x

) − λ

n−1

µ

nP ∗ n−1

(

x

),

n

>

0

,

but P₁∗

(

x

) =

x

−

λ

₀

−

c

=

P1

(

x

) −

c

.

Evidently, there is unique (natural) orthogonalizing measure

ψ

∗for the polynomials P_n∗and we can define quantities

ξ

_i∗and

σ

∗_{in terms of}

_ψ

∗_{analogously to(3)–(5). Moreover}

_ξ

∗

i is the limit as n

→ ∞

of x ∗

ni, the ith smallest zero of the polynomial

P∗ n

(

x

)

.

Given the polynomials Pn, the polynomials Pn∗are called co-recursive polynomials and have been studied for the first time by Chihara [15]. In particular, applying [15, Theorem 1] to the situation at hand, we have

xn,i

<

x∗n,i

<

xn,i+1

<

x∗n,i+1 i

=

1

, . . . ,

n

−

1

,

n

>

0

.

Subsequently letting n tend to infinity we obtain

ξ

i

≤

ξ

i∗

≤

ξ

i+1

≤

ξ

i∗+1 i

≥

1

,

(18)

and hence

σ

∗

₌

_{σ .}

(19) We have now gathered sufficient information to conclude that statement (i) ofCorollary 1implies statement (ii). Indeed, suppose the parameters in the recurrence relation for the polynomials Pnsatisfy K∞

< ∞

and

µ

0

=

0. Then, by applying

Corollary 1(i) to the polynomials P∗

n we conclude that

ξ

∗ 1

>

0 is equivalent to

σ

∗

_>

_{0, and to sup} nLn

(

K∞

−

Kn

) < ∞

. But

ξ

∗

1

>

0 is equivalent to

ξ

2

>

0 since

ξ

1

≤

ξ

1∗

≤

ξ

2

≤

ξ

2∗, by(18), while we cannot have

ξ

∗ 1

=

0 if

ξ

∗ 2

>

0, by(12). Finally,

σ

∗

_>

0 is equivalent to

σ >

0 by(19).

In view of(19)it also follows that to proveCorollary 2(i) it suffices to establish the result for

µ

0

>

0.

4.3. Associated polynomials

Throughout this subsection we assume K∞

< ∞

. The associated (or numerator) polynomials Pn(k)of order k

≥

0 associated with the sequence

{

Pn

}

of(1)are given by the recurrence relation

P_n(k₊)₁

(

x

) = (

x

−

λ

_n+k

−

µ

n+k

)

Pn(k)

(

x

) − λ

n+k−1

µ

n+kPn(k−)1

(

x

),

n

>

0

,

P₁(k)

(

x

) =

x

−

λ

_k

−

µ

_k

,

P₀(k)

(

x

) =

1

.

Defining

ξ

_i(k)and

σ

(k)as in(3)–(5)with

ψ

replaced by

ψ

(k)we have

ξ

(k) 1

≤

ξ

(k+1) 1

,

k

≥

0

,

and _klim_→∞

ξ

(k) 1

=

σ

(20)

from [3, Theorem III.4.2] and [16, Theorem 1], respectively. Moreover, defining

π

n(k), Kn(k), K∞(k)and L(nk)as in(7)–(9)with

λ

n and

µ

nreplaced by

λ

n+kand

µ

n+k, respectively, it is readily seen that

π

i(k)

=

π

i+k

/π

k, so that

K_n(k)

=

1

π

k

(

Kn+k

−

Kk−1

) ,

K∞(k)

=

1

π

k

(

K∞

−

Kk−1

) < ∞,

and L(_nk)

=

π

_k

(

Ln+k

−

Lk−1

).

(These relations are valid for k

≥

0 if we let K−1

=

L−1

=

0.) It follows that R(k)

:=

supnL( k)

n

(

K∞(k)

−

Kn(k)

)

satisfies

R(k)

=

sup n

(

Ln+k

−

Lk−1

)(

K∞

−

Kn+k

).

ApplyingProposition 2to

ξ

₁(k)we find that

1 4R(k)

≤

ξ

(k) 1

≤

1 R(k)

,

k

≥

0

,

so by(20)we have

σ = ∞

if and only if limk→∞R(k)

= ∞

, which is easily seen to be equivalent to statement (i) of Corollary 2.

4.4. Proposition 2: Sketch of proof and remarks

The zeros xniof the polynomials Pnof(1)may be interpreted as eigenvalues of a symmetric tridiagonal matrix (or Jacobi

matrix). Indeed, let I denote the identity matrix and

Jn

:=



















λ

0

+

µ

0

−



λ

0

µ

1 0

· · ·

0 0

−



λ

₀

µ

₁

λ

₁

+

µ

₁

−



λ

₁

µ

₂

· · ·

0 0 0

−



λ

₁

µ

₂

λ

₂

+

µ

₂

· · ·

0 0

...

...

...

...

...

...

0 0 0

· · ·

λ

_n−1

+

µ

n−1

−



λ

n−1

µ

n 0 0 0

· · ·

−



λ

_n−1

µ

n

λ

n

+

µ

n



















.

Then, expanding det

(

xI

−

Jn

)

by its last row and comparing the result with the recurrence relation(1), it follows that we can identify det

(

xI

−

Jn

)

with the polynomial Pn+1

(

x

)

. So a representation for

ξ

1

=

limn→∞xn1may be obtained by letting n tend to infinity in a representation of the smallest eigenvalue of the Jacobi matrix Jn. The latter may be obtained by minimizing the Rayleigh quotient

R

(

Jn

,

x

) :=

xT_J

nx xT_x

of Jnover all nonzero vectors x (see, for example, [17, Section 7.5]). Actually, precisely this approach was adopted in [18, Section 5] to get representations for xn1and

ξ

1. However, to proveProposition 2a subtler approach is needed. Namely,

replacing Jnby

˜

Jn

:=



















λ

0

+

µ

0

−



λ

0

µ

1 0

· · ·

0 0

−



λ

₀

µ

₁

λ

₁

+

µ

₁

−



λ

₁

µ

₂

· · ·

0 0 0

−



λ

₁

µ

₂

λ

₂

+

µ

₂

· · ·

0 0

...

...

...

...

...

...

0 0 0

· · ·

λ

_n−1

+

µ

n−1

−



λ

n−1

µ

n 0 0 0

· · ·

−



λ

_n−1

µ

n

µ

n



















,

the polynomialsP

˜

n+1

(

x

) :=

det

(

xI

− ˜

Jn

)

are readily seen to satisfy

˜

Pn+1

(

x

) =

Pn+1

(

x

) + λ

nPn

(

x

),

n

≥

0

,

and can therefore be identified as quasi-orthogonal polynomials (see [3, Section II.5]). As a consequenceP

˜

n

(

x

)

has real and simple zeros

˜

xn1

< ˜

xn2

< · · · < ˜

xnn, which are separated by the zeros of Pn

(

x

)

. Moreover, it is not difficult to verify that

˜

xn1

<

xn1, so that

ξ

˜

1

:=

limn→∞x

˜

n1

≤

ξ

1. But, seeing(15), the polynomialsP

˜

ncan also be identified with the dual polynomials

Pd

nintroduced in Section4.1. So it follows with(12)and(16)that in the setting at hand we actually have

ξ

˜

1

=

ξ

1d

=

ξ

1. To get

a representation for

ξ

1we may therefore start with the representation forx

˜

n1obtained by minimizing the Rayleigh quotient of

˜

_J

nand subsequently let n tend to infinity. Proceeding in this way leads to the representation

ξ

1

=

inf x



























∞



i=0

µ

i

π

ix2i ∞



i=0

π

i



i



j=0 xj



2



























,

(21)

where x

=

(

x0

,

x1

, . . .)

is an infinite sequence of real numbers with finitely many nonzero elements.Proposition 2emerges

after applying the weighted discrete Hardy’s inequalities given in [19]. For the details of the proof we refer to [12].

The results in [12] include representations in the spirit of(21)for the decay parameter of a birth–death process under all possible scenarios. The proofs of these results require a representation for the second smallest eigenvalue of a Jacobi matrix, which is obtained in [12] by applying the Courant–Fischer theorem, an extension of the method involving Rayleigh quotients used above to represent the smallest eigenvalue. Being content in this paper with criteria for positivity rather than representations, there is no need to appeal to the full Courant–Fischer theorem.

5. Related literature and concluding remarks

We have noted in the introduction that in the setting of birth–death processes it is of particular interest to be able to establish whether the transition probabilities converge to their limiting values exponentially fast. In view of(6)this question

may be translated in the current setting by asking whether 0

< σ ≤ ∞

, soCorollary 1provides us with a simple means to check whether the decay parameter of a birth–death process is positive.

In the orthogonal-polynomial literature the question of whether the support of an orthogonalizing measure is discrete with no finite limit point has received some attention, notably in the work of Chihara (see [3, Chapter IV], [20–22]). Chihara’s point of departure usually is the three-terms recurrence relation

Pn+1

(

x

) = (

x

−

cn

)

Pn

(

x

) − ρ

nPn−1

(

x

),

n

>

0

,

P1

(

x

) =

x

−

c0

,

P0

(

x

) =

1

,

(22) where

ρ

n

>

0. Note that we regain the polynomials Pnof(1)if

cn

=

λ

n

+

µ

n

,

ρ

n+1

=

λ

n

µ

n+1

,

n

≥

0

.

(23)

Interestingly, by [3, Corollary to Theorem I.9.1] the existence of positive numbers

λ

nand

µ

n(except

µ

0

≥

0) satisfying(23)

is not only sufficient, but also necessary for the Stieltjes moment problem associated with the polynomials

{

Pn

}

of(22)to be solvable. Moreover, if such numbers exist one can always choose

µ

0

=

0. So in view ofCorollary 2, and considering that the

sequence

{

Pn

}

is orthogonal with respect to a measure on

[

a

, ∞)

if and only if the sequence

{

Qn

}

, with Qn

(

x

) :=

Pn

(

x

−

a

)

, is orthogonal with respect to a measure on

[

0

, ∞)

, we can formulate the following result with regard to the polynomials(22).

Proposition 3. The polynomials

{

Pn

}

of (22)are orthogonal with respect to a measure on the interval

[

a

, ∞)

with discrete

support and no finite limit point if and only if the numbers

λ

nand

µ

nrecurrently defined by

λ

0

:=

c0

−

a and

µ

n

:=

ρ

n+1

/λ

n

, λ

n

:=

cn

−

a

−

µ

n

,

n

=

1

,

2

, . . . ,

are all positive and – using the notation(7)–(9)– satisfy Ln

(

K∞

−

Kn

) →

0 or Kn

(

L∞

−

Ln

) →

0 as n

→ ∞

.

The question of whether

σ = ∞

in the specific setting of birth–death polynomials has been addressed by Chihara in [22], and earlier by Lederman and Reuter [23], Maki [24] and the present author [6]. By [3, Theorem IV.3.1] a necessary condition for

σ = ∞

is cn

→ ∞

, so an interesting case arises in the birth–death setting when

λ

n

=

anα

+

o

(

nα

),

µ

n

=

bnβ

+

o

(

nβ

),

n

≥

0

,

(24) where a

,

b

, α, β

are nonnegative constants such that

µ

0

≥

0 and

λ

n

>

0,

µ

n+1

>

0 for n

≥

0. By employing a criterion

involving chain sequences Chihara [22] proves that

σ = ∞

if

α ̸= β

, or if

α = β

but a

̸=

b, a conclusion that may be reached

also by applyingCorollary 2. Chihara demonstrates in addition that both

σ = ∞

and

σ < ∞

may occur if

α = β

, a

=

b and

α ≤

2, thus refuting the conjecture in [25] that the spectrum in this case is continuous. Chihara suspects the claim in [25], that always

σ = ∞

when

α = β

, a

=

b and

α >

2, to be true, but he can verify it only under additional assumptions on the rates. But actually,

σ

may be finite for all

α >

0, as the following example shows. Let

λ

0

=

1

, µ

0

=

0 and

λ

n

=

nα

, µ

n

=

nα

(

1

+

gn

),

n

>

0

,

where, for k

=

0

,

1

, . . .

, gn

=











1 2k

+

1 n

=

n2k

+

1

, . . . ,

n2k+1

−

1 2k

+

2 n

=

n2k+1

+

1

, . . . ,

n2k+2

,

with n0

=

0 and n1

<

n2

< · · ·

successively chosen such that

Gn2k+1

>

1 and n α 2k+2Gn2k+2

<

1

,

k

=

0

,

1

, . . . ,

where Gn

=

n



i=1

(

1

+

gi

),

n

≥

1

.

Since

π

n

=

(

nαGn

)

−1

,

(λ

n

π

n

)

−1

=

Gn

,

it follows that K∞

=

L∞

= ∞

. So byProposition 1we have

σ =

0. We conclude this section with the following observation. Letting

C

:=

∞



n=0

(λ

n

π

n

)

−1 Kn and D

:=

∞



n=0

(λ

n

π

n

)

−1

₍

K∞

−

Kn

),

it is shown in [26, Theorem 2] that

C

< ∞

or D

< ∞ ⇐⇒



i>1

1

ξ

i

< ∞,

whence

σ = ∞

if C

< ∞

or D

< ∞

, a conclusion that may be drawn also from the main theorem in [27]. But, with K−1

=

L−1

=

0, we actually have C

=

∞



n=0

(λ

n

π

n

)

−1Kn

=

∞



n=0

π

n

(

L∞

−

Ln−1

)

=

∞



n=0

(

Kn

−

Kn−1

)(

L∞

−

Ln−1

)

=

∞



n=0



Kn

(

L∞

−

Ln

) −

Kn−1

(

L∞

−

Ln−1

) + (λ

n

π

n

)

−1Kn



=

lim n→∞Kn

(

L∞

−

Ln

) +

C

,

so that C

< ∞ H⇒

lim n→∞Kn

(

L∞

−

Ln

) =

0

.

Similarly, D

< ∞ H⇒

lim n→∞Ln

(

K∞

−

Kn

) =

0

.

So the fact that

σ = ∞

if C

< ∞

or D

< ∞

can be concluded fromCorollary 2as well. Note that our assumption(10)is equivalent to C

+

D

= ∞

.

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